If a company cuts ties with a real estate agent, does that mean that he or she also loses access to the MLS?

They lose access until they find another broker who is a member of the local MLS who will hold their license. They cannot legally work without such an association.

'Anonymous' is wrong. Agents do NOT belong to the local real estate board/MLS.  They have an ASSOCIATE membership, which is ONLY valid as long as their license is held by the member broker. 

An agent must be working for a broker to have access to the MLS.  So yes, the agent loses access to the MLS without being affilated (working under the license of a broker).  

Also an agent must be under a broker license in order to beable to take a listing, write an offer or get paid for any real estate function.

No.  The agent is usually a member of the local board of realtors who administers the local MLS, thus continues to have access, but can't do business until they have established themselves through another broker (company) or become a broker themselves (establishing their own company).

No. You can access the MLS online. However there are going to be some listings that only a broker will be able to see.